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Friday, 8 February 2019

Answer to A Beautiful Mind puzzle

Two trains are on the same track. They start 100km apart and head towards each other at a speed of 50km/h.

Whilst
these two trains are heading for their collision a fly starts out on
the front of one train and zooms directly to the front of the other at a
speed of 75km/h (see the animation above). Once the fly reaches the
second train it immediately darts back to the front of the first train
at the same speed and repeats this back and forth motion until the two
trains collide and the fly is squashed on impact.

How far has the fly traveled, before it meets its demise?

One way to approach this problem is through infinite series. Namely, we find how far the fly during the first journey, the second journey, the third journey, etc. and add them all up. Thankfully, there is a fairly nice formula that provides this answer.

However, a much simpler way to calculate the distance is by realising that the changes in direction do not matter. Namely, all we are asking is how far can a fly travel in the hour it takes for the trains to hit each other? Clearly, this is simply 75 km. Sometimes, a moment's thought can save an hour's work!

As mentioned last time, John von Neumann was said to have immediately answered this problem, but when pressed on his solution method he said that he has used the infinite series method. Ah to have the mind of a genius!

This and other aspects of von Neumann's genius are discussed in Raymond Flood's excellent Gresham College talk, below (plus you get a bit of Alan Turing for free, which Thomas is always happy about).